First let's review the geometric origin [1]} of the logarithmic factor in the scrambling time for black holesThis section is a summary of the more technical discussion in [2]}[3]}[4]}[5]}.
The holographic degrees of freedom in AdS /CFT are located at the boundary of AdS but it is often convenient to regulate the theory so that the effective degrees of freedom are closer in. For example we can think of them as being located on a shell about one AdS length from the horizon at the edge of the near-horizon region. For a black hole in flat space a convenient location is at the photon sphere where near-horizon “zone" gives way to the Newtonian region.
